Passing geekish, yes.
In the book, when the Nine were initially looking for the Ring they were trying to be (relatively) inconspicuous. [Not like in the movie, obviously.] Flying on wyrmback is just about the polar opposite of inconspicuous.
Agree re the Eye in the movie. That flashlight-beam it emitted suggested that it worked line-of-sight which the books (and other parts of the movies) contradicted.
Questions of the Ring corrupting its bearer aside, "Why not have an eagle carry the Ring" raises another question; namely, "Why weren't the eagles doing 24/7 reconaissance on Mordor, or even bombing runs?" [Gandalf would have thought of it, if no one else.] That they weren't suggests to me that Sauron had some pretty effective air defenses.
Which was probably the single largest logical flaw in the book. The Nine of the Fellowship part of the book, sneaking into The Prancing Pony and fleeing in terror from Aragorn and a few torches held by hobbits, have almost nothing in common with the Nine of ROTK, who go toe to toe with Gandalf the White, with the strong implication that Gandalf might be defeated by the Witch-King, although the confrontation never quite materializes in the book.
If the Nine are so powerful, why are they sneaking around trying to grab the Ring by stealth? Why not take it by force?